he paraboloid
z = 4 − x − x2 − 2y2
intersects the plane x = 4 in a parabola. Find parametric equations in terms of t for the tangent line to this parabola at the point
(4, 2, −24).
(Enter your answer as a comma-separated list of equations. Let x, y, and z be in terms of t.)
x = 4, y = 2 + t, z = -24 + 8t
To find the parametric equations for the tangent line to the parabola at the point (4, 2, -24), we first need to find the equation of the parabola.
Given that the paraboloid intersects the plane x = 4, we can substitute x = 4 into the equation of the paraboloid:
z = 4 - x - x^2 - 2y^2
z = 4 - 4 - 4^2 - 2y^2
z = -16 - 2y^2
Now, we want to find the values of y and z at the point (4, 2, -24). Substituting x = 4 into the equation above:
-24 = -16 - 2y^2
-8 = -2y^2
4 = y^2
y = ±2
Thus, the parabola intersects the plane x = 4 at two points: (4, 2, -24) and (4, -2, -24).
To find the tangent line at the point (4, 2, -24), we'll differentiate the equation of the parabola with respect to x:
dz/dx = -1
Now, we can find the slope of the tangent line at the point (4, 2, -24) using the derivative:
m = dz/dx = -1
Thus, the slope of the tangent line is -1.
To find the parametric equations for the tangent line, we'll start with the point-slope form:
z - z1 = m(x - x1)
Substituting the values (x1, y1, z1) = (4, 2, -24) and m = -1:
z - (-24) = -1(x - 4)
z + 24 = -x + 4
z = -x - 20
Therefore, the parametric equations in terms of t for the tangent line to the parabola at the point (4, 2, -24) are:
x = t + 4
y = 2
z = -t - 20
Therefore, the parametric equations for the tangent line are:
x = t + 4, y = 2, z = -t - 20
To find the parametric equations for the tangent line to the parabola at the given point, we need to find the derivative of the paraboloid equation with respect to t and evaluate it at the given point.
1. Differentiate the given equation, z = 4 − x − x^2 − 2y^2, with respect to t:
dz/dt = -dx/dt - 2y * dy/dt
2. Since we are given the point (4, 2, -24), plug in these values into the equation:
-24 = -dx/dt - 2(2) * dy/dt
3. Solve the equation for -dx/dt and dy/dt:
-24 + 4dy/dt = -dx/dt
4. Now we have two derivatives, -dx/dt and dy/dt, expressed in terms of t:
-24 + 4dy/dt = -dx/dt
5. The parametric equations for the tangent line can be written in the form:
x = 4 + at
y = 2 + bt
z = -24 + ct
6. Plug in the expressions for -dx/dt and dy/dt into the parametric equations:
x = 4 - 24t
y = 2 + 4t
z = -24 + ct
Therefore, the parametric equations for the tangent line to the parabola at the point (4, 2, -24) are:
x = 4 - 24t
y = 2 + 4t
z = -24 + ct